(45,12,3) block plan
The (45,12,3) block diagram is a special symmetrical block diagram . In order to be able to construct it, this combinatorial problem had to be solved: An empty 45 × 45 matrix was filled with ones in such a way that each row of the matrix contains exactly 12 ones and any two rows have exactly 3 ones in the same column (not more and not less). That sounds relatively simple, but it is not trivial to solve. There are only certain combinations of parameters (like here v = 45, k = 12, λ = 3) for which such a construction is feasible. The smallest of these (v, k, λ) are listed in this overview .
designation
This symmetrical 2- (45,12,3) block diagram is called a 9th order triplane .
properties
This symmetrical block diagram has the parameters v = 45, k = 12, λ = 3 and thus the following properties:
- It consists of 45 blocks and 45 points.
- Each block contains exactly 12 points.
- Every 2 blocks intersect in exactly 3 points.
- Each point lies on exactly 12 blocks.
- Each 2 points are connected by exactly 3 blocks.
Existence and characterization
There are at least 3752 non-isomorphic 2- (45,12,3) block plans. One of these solutions is:
- Solution 1 with the signature 12 x 1, 6 x 2, 4 x 3, 12 x 4, 6 x 11, 1 x 12, 4 x 15. It contains 1140 ovals of the 4th order.
List of blocks
All the blocks of this block plan are listed here; See this illustration to understand this list
- Solution 1
2 3 4 5 6 7 8 9 10 11 12 13 1 3 4 5 14 15 16 17 18 19 20 21 1 2 4 5 22 23 24 25 26 27 28 29 1 2 3 5 30 31 32 33 34 35 36 37 1 2 3 4 38 39 40 41 42 43 44 45 1 7 8 9 14 15 22 23 30 31 38 39 1 6 8 9 16 17 24 25 32 33 40 41 1 6 7 9 18 19 26 27 34 35 42 43 1 6 7 8 20 21 28 29 36 37 44 45 1 11 12 13 14 15 24 25 34 35 44 45 1 10 12 13 16 17 22 23 36 37 42 43 1 10 11 13 18 19 28 29 30 31 40 41 1 10 11 12 20 21 26 27 32 33 38 39 2 6 10 15 16 18 22 28 32 34 38 44 2 6 10 14 17 19 23 29 33 35 39 45 2 7 11 14 18 20 23 24 32 36 40 42 2 7 11 15 19 21 22 25 33 37 41 43 2 8 12 14 16 20 26 28 30 35 41 43 2 8 12 15 17 21 27 29 31 34 40 42 2 9 13 16 18 21 24 27 30 37 39 45 2 9 13 17 19 20 25 26 31 36 38 44 3 6 11 14 17 24 27 28 31 37 38 43 3 6 11 15 16 25 26 29 30 36 39 42 3 7 10 16 20 22 25 27 31 35 40 45 3 7 10 17 21 23 24 26 30 34 41 44 3 8 13 18 21 23 25 28 33 35 38 42 3 8 13 19 20 22 24 29 32 34 39 43 3 9 12 14 18 22 26 29 33 37 40 44 3 9 12 15 19 23 27 28 32 36 41 45 4 6 12 18 20 23 25 31 34 37 39 41 4 6 12 19 21 22 24 30 35 36 38 40 4 7 13 14 16 27 29 33 34 36 38 41 4 7 13 15 17 26 28 32 35 37 39 40 4 8 10 14 19 25 27 30 32 37 42 44 4 8 10 15 18 24 26 31 33 36 43 45 4 9 11 16 21 23 29 31 32 35 43 44 4 9 11 17 20 22 28 30 33 34 42 45 5 6 13 14 21 22 26 31 32 41 42 45 5 6 13 15 20 23 27 30 33 40 43 44 5 7 12 16 19 24 28 31 33 39 42 44 5 7 12 17 18 25 29 30 32 38 43 45 5 8 11 16 19 23 26 34 37 38 40 45 5 8 11 17 18 22 27 35 36 39 41 44 5 9 10 14 21 25 28 34 36 39 40 43 5 9 10 15 20 24 29 35 37 38 41 42
Incidence matrix
This is a representation of the incidence matrix of this block diagram; see this illustration to understand this matrix
- Solution 1
. O O O O O O O O O O O O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O . O O O . . . . . . . . O O O O O O O O . . . . . . . . . . . . . . . . . . . . . . . . O O . O O . . . . . . . . . . . . . . . . O O O O O O O O . . . . . . . . . . . . . . . . O O O . O . . . . . . . . . . . . . . . . . . . . . . . . O O O O O O O O . . . . . . . . O O O O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O O O O O O O O O . . . . . O O O . . . . O O . . . . . . O O . . . . . . O O . . . . . . O O . . . . . . O . . . . O . O O . . . . . . O O . . . . . . O O . . . . . . O O . . . . . . O O . . . . O . . . . O O . O . . . . . . . . O O . . . . . . O O . . . . . . O O . . . . . . O O . . O . . . . O O O . . . . . . . . . . . O O . . . . . . O O . . . . . . O O . . . . . . O O O . . . . . . . . . O O O O O . . . . . . . . O O . . . . . . . . O O . . . . . . . . O O O . . . . . . . . O . O O . . O O . . . . O O . . . . . . . . . . . . O O . . . . O O . . O . . . . . . . . O O . O . . . . O O . . . . . . . . O O O O . . . . . . . . O O . . . . O . . . . . . . . O O O . . . . . . . O O . . . . O O . . . . O O . . . . O O . . . . . . . O . . . O . . . O . . . . O O . O . . . O . . . . . O . . . O . O . . . O . . . . . O . . O . . . O . . . O . . . O . . O . O . . . O . . . . . O . . . O . O . . . O . . . . . O . O . . . . O . . . O . . O . . . O . O . . O O . . . . . . . O . . . O . . . O . O . . . . O . . . . O . . . O . . . O . . . O . O O . . O . . . . . . . O . . . O . . . O . O . . . O . . . . . O . . . O . O . O . . . O . . . . . O . O . O . . . . O . . . . . O . O . . . O . . . . . O . . . O . . O . O . . . O . . . . . O . O . O . . O . . . . . O . O . . . . O . . . . . . O . . . O . . O . O . . O . . O . . O . . O . . . . . . O . O . . . . . O . O . . . . . . O . . . O . . . O . O O . . . . O O . . . . O . . . . O . O . . . . . O . . . O . . O . . . . O . . O . . O . . . . . . O . . O O . . O . . . . . O O . . . . O . . . . O . . O . . . . O . . . O O . . . . . . . . O O . . O O . . . . . O . . O . . O . . . . . O . . . O . . O . . . . . O . . . O . O . . O . O . . . O . . . O . . . . O . . . . O . . O . . . O . . O . . . . . . O . . . O . O O . O . . . O . . . O . . . . . . O . . O . . . O . . . . O . . . . O . . . . O . . O . O . O . . O . . . . O . O . . O . . . O . . . . . O . . . . O . . . . O . . . . . O O . O . O . . . . O . . O . O . . . . O . . . O . . . . O . . . . . O . . O . O . . . O . . . O . . . O . . O . . . O . . . O . . O . . . O . . . O . . . . . O . . O . . O . . . O . . . O . . . O O . . . O . . . O . . . . O . . . O . . . O . O . . . . . O . . . . . O . O . . O . O . . . . . O . . O . . O . O . O . . . . . . . O . O . . . . . O . . . . . . O . O O . O . . . . . O . . . . O O . O . O . . . . . . . . O . . O . . . . . O O . O . . . . . . . . . . O . O . . . O O . O . O . . O . . . . . . . O . . O . . . . . O . O . O . . . . . . . . O . O . . . O . . O . O . O O . . . . . . . . O . . . O . O . . . O . . . . O . . . . . O . O . . O . O . . . . O . . . . O . O . . . . O . . . O . O . . . . O . . O . . . . . O . O . . . . O . O . . O . . . . . . O . O . . . O . . . . O . O . . . . O . . . . O . O . . . . . O . O O . . O . . . . . . . O O . . . . O . . . . O . O . . . . . O . . O . O . . . . . O . O . . O O . . . . . . . O . . O . . . . O O . . . . . . O O . . . . . . O O . . . O . . . . O O . . . . . . . . O O . . O . . . . O O . . . . . . O . O . . . . O . . O . . . O . . O . . O . . . . . . O . . O O . . . . . O . O . . . . O . . . O . . O . . . . O . . . O . . O . O . . . . . O . . O . O . . . . . O . O . . . . O . . . . O O . . . . . . O . . . O O . O . . . . . O . . . . O . O . . . . O . . O . . O . . . . O . . O . . . O . . O . . . . . . . O . . O O . O . . . . O . . . . O . . O . . O . . . . . O O . . . O . . . . O . . . . . . . O O . . O . O . . O . . . . . O . . . O O . . . O . . . . . . O . . . O . . O . . . . . O . O . . O O . . O . . . . . . O . . . O O . . . . O . . . . O . . . O . . . . O . . . . . O . O O . . O O . . .
oval
An oval of the block plan is a set of its points, no three of which are on a block. Here is an example of a maximum order oval from this block diagram:
- Solution 1
1 2 7 16
literature
- Thomas Beth , Dieter Jungnickel , Hanfried Lenz : Design Theory . 1st edition. BI Wissenschaftsverlag, Mannheim / Vienna / Zurich 1985, ISBN 3-411-01675-2 .
- Albrecht Beutelspacher : Introduction to Finite Geometry. Volume 1: Block Plans . BI Wissenschaftsverlag, Mannheim / Vienna / Zurich 1982, ISBN 3-411-01632-9 .
Individual evidence
- ^ Rudolf Mathon, Alexander Rosa : 2- (ν, κ, λ) Designs of Small Order. In: Charles J. Colbourn , Jeffrey H. Dinitz (Eds.): Handbook of Combinatorial Designs. 2nd edition. Chapman and Hall / CRC, Boca Raton FL et al. 2007, ISBN 978-1-4200-1054-1 , pp. 25-57.